The escape velocity v of a body depends on- (i) the acceleration due to gravity ‘g’ of the planet, (ii) the radius R of the planet. Establish dimensionally the relation for the escape velocity.

v a gᵅ Rᵇ ⇒ v = k gᵅ Rᵇ, K → dimensionless proportionality constant
[V] = [g]ᵅ [R]ᵇ
[M⁰L¹T⁻¹] = [M⁰L¹T⁻²] [M⁰L¹T⁰]ᵇ
equating powers
1 = a + b
– 1 = – 2a ⇒ a= 1/2
b= 1 – a = 1-1/2 = 1/2
v=k √gR